Problem of the Month (July 2006)

In the chess literature, the problem of placing the minimum number of pieces to attack every square of the chessboard is well known. This month we consider a natural extension, placing pieces on a board so that each square is attacked an equal number of times. That is, what is the minimum number of white chess pieces that can be placed on an n×n chessboard so that each vacant square is attacked exactly k times? What if we want to attack every square exactly k times? (We allow any number of pieces, in any location, so pawns can be on the first or last rank.)

Can you find solutions with few pieces for n=3? What about larger square boards? What about rectangular boards? Are the answers different if you are allowed to use both black and white pieces?

ANSWERS

Let V(n,k) be the fewest number of chess pieces needed so that each vacant square of an n×n chessboard is attacked exactly k times.